A complete and statistically consistent uncertainty quantification for deep learning is provided, including the sources of uncertainty arising from (1) the new input data, (2) the training and testing data (3) the weight vectors of the neural network, and (4) the neural network because it is not a perfect predictor. Using Bayes Theorem and conditional probability densities, we demonstrate how each uncertainty source can be systematically quantified. We also introduce a fast and practical way to incorporate and combine all sources of errors for the first time. For illustration, the new method is applied to quantify errors in cloud autoconversion rates, predicted from an artificial neural network that was trained by aircraft cloud probe measurements in the Azores and the stochastic collection equation formulated as a two-moment bin model. For this specific example, the output uncertainty arising from uncertainty in the training and testing data is dominant, followed by uncertainty in the input data, in the trained neural network, and uncertainty in the weights. We discuss the usefulness of the methodology for machine learning practice, and how, through inclusion of uncertainty in the training data, the new methodology is less sensitive to input data that falls outside of the training data set.

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